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Theorem infdifsn 9272
Description: Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
infdifsn (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)

Proof of Theorem infdifsn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 8639 . . . 4 (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1𝐴)
21adantr 484 . . 3 ((ω ≼ 𝐴𝐵𝐴) → ∃𝑓 𝑓:ω–1-1𝐴)
3 reldom 8632 . . . . . . 7 Rel ≼
43brrelex2i 5606 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
54ad2antrr 726 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐴 ∈ V)
6 simplr 769 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐵𝐴)
7 f1f 6615 . . . . . . 7 (𝑓:ω–1-1𝐴𝑓:ω⟶𝐴)
87adantl 485 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω⟶𝐴)
9 peano1 7667 . . . . . 6 ∅ ∈ ω
10 ffvelrn 6902 . . . . . 6 ((𝑓:ω⟶𝐴 ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ 𝐴)
118, 9, 10sylancl 589 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓‘∅) ∈ 𝐴)
12 difsnen 8727 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝐴 ∧ (𝑓‘∅) ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
135, 6, 11, 12syl3anc 1373 . . . 4 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
14 vex 3412 . . . . . . . . . 10 𝑓 ∈ V
15 f1f1orn 6672 . . . . . . . . . . 11 (𝑓:ω–1-1𝐴𝑓:ω–1-1-onto→ran 𝑓)
1615adantl 485 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω–1-1-onto→ran 𝑓)
17 f1oen3g 8644 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:ω–1-1-onto→ran 𝑓) → ω ≈ ran 𝑓)
1814, 16, 17sylancr 590 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ ran 𝑓)
1918ensymd 8679 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓 ≈ ω)
203brrelex1i 5605 . . . . . . . . . . 11 (ω ≼ 𝐴 → ω ∈ V)
2120ad2antrr 726 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ∈ V)
22 limom 7660 . . . . . . . . . . 11 Lim ω
2322limenpsi 8821 . . . . . . . . . 10 (ω ∈ V → ω ≈ (ω ∖ {∅}))
2421, 23syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ (ω ∖ {∅}))
2514resex 5899 . . . . . . . . . . 11 (𝑓 ↾ (ω ∖ {∅})) ∈ V
26 simpr 488 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω–1-1𝐴)
27 difss 4046 . . . . . . . . . . . 12 (ω ∖ {∅}) ⊆ ω
28 f1ores 6675 . . . . . . . . . . . 12 ((𝑓:ω–1-1𝐴 ∧ (ω ∖ {∅}) ⊆ ω) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
2926, 27, 28sylancl 589 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
30 f1oen3g 8644 . . . . . . . . . . 11 (((𝑓 ↾ (ω ∖ {∅})) ∈ V ∧ (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅}))) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
3125, 29, 30sylancr 590 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
32 f1orn 6671 . . . . . . . . . . . . 13 (𝑓:ω–1-1-onto→ran 𝑓 ↔ (𝑓 Fn ω ∧ Fun 𝑓))
3332simprbi 500 . . . . . . . . . . . 12 (𝑓:ω–1-1-onto→ran 𝑓 → Fun 𝑓)
34 imadif 6464 . . . . . . . . . . . 12 (Fun 𝑓 → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
3516, 33, 343syl 18 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
36 f1fn 6616 . . . . . . . . . . . . . 14 (𝑓:ω–1-1𝐴𝑓 Fn ω)
3736adantl 485 . . . . . . . . . . . . 13 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓 Fn ω)
38 fnima 6508 . . . . . . . . . . . . 13 (𝑓 Fn ω → (𝑓 “ ω) = ran 𝑓)
3937, 38syl 17 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ ω) = ran 𝑓)
40 fnsnfv 6790 . . . . . . . . . . . . . 14 ((𝑓 Fn ω ∧ ∅ ∈ ω) → {(𝑓‘∅)} = (𝑓 “ {∅}))
4137, 9, 40sylancl 589 . . . . . . . . . . . . 13 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → {(𝑓‘∅)} = (𝑓 “ {∅}))
4241eqcomd 2743 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ {∅}) = {(𝑓‘∅)})
4339, 42difeq12d 4038 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝑓 “ ω) ∖ (𝑓 “ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
4435, 43eqtrd 2777 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ (ω ∖ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
4531, 44breqtrd 5079 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
46 entr 8680 . . . . . . . . 9 ((ω ≈ (ω ∖ {∅}) ∧ (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
4724, 45, 46syl2anc 587 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
48 entr 8680 . . . . . . . 8 ((ran 𝑓 ≈ ω ∧ ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
4919, 47, 48syl2anc 587 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
50 difexg 5220 . . . . . . . 8 (𝐴 ∈ V → (𝐴 ∖ ran 𝑓) ∈ V)
51 enrefg 8660 . . . . . . . 8 ((𝐴 ∖ ran 𝑓) ∈ V → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
525, 50, 513syl 18 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
53 disjdif 4386 . . . . . . . 8 (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅
5453a1i 11 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅)
55 difss 4046 . . . . . . . . . 10 (ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓
56 ssrin 4148 . . . . . . . . . 10 ((ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓 → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)))
5755, 56ax-mp 5 . . . . . . . . 9 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓))
58 sseq0 4314 . . . . . . . . 9 ((((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
5957, 53, 58mp2an 692 . . . . . . . 8 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅
6059a1i 11 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
61 unen 8723 . . . . . . 7 (((ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓)) ∧ ((ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅ ∧ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
6249, 52, 54, 60, 61syl22anc 839 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
638frnd 6553 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓𝐴)
64 undif 4396 . . . . . . 7 (ran 𝑓𝐴 ↔ (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
6563, 64sylib 221 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
66 uncom 4067 . . . . . . 7 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)}))
67 eldifn 4042 . . . . . . . . . . 11 ((𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓) → ¬ (𝑓‘∅) ∈ ran 𝑓)
68 fnfvelrn 6901 . . . . . . . . . . . 12 ((𝑓 Fn ω ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ ran 𝑓)
6937, 9, 68sylancl 589 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓‘∅) ∈ ran 𝑓)
7067, 69nsyl3 140 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
71 disjsn 4627 . . . . . . . . . 10 (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ ↔ ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
7270, 71sylibr 237 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅)
73 undif4 4381 . . . . . . . . 9 (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
7472, 73syl 17 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
75 uncom 4067 . . . . . . . . . 10 ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓))
7675, 65eqtrid 2789 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = 𝐴)
7776difeq1d 4036 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}) = (𝐴 ∖ {(𝑓‘∅)}))
7874, 77eqtrd 2777 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (𝐴 ∖ {(𝑓‘∅)}))
7966, 78eqtrid 2789 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = (𝐴 ∖ {(𝑓‘∅)}))
8062, 65, 793brtr3d 5084 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐴 ≈ (𝐴 ∖ {(𝑓‘∅)}))
8180ensymd 8679 . . . 4 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴)
82 entr 8680 . . . 4 (((𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
8313, 81, 82syl2anc 587 . . 3 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
842, 83exlimddv 1943 . 2 ((ω ≼ 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
85 difsn 4711 . . . 4 𝐵𝐴 → (𝐴 ∖ {𝐵}) = 𝐴)
8685adantl 485 . . 3 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → (𝐴 ∖ {𝐵}) = 𝐴)
87 enrefg 8660 . . . . 5 (𝐴 ∈ V → 𝐴𝐴)
884, 87syl 17 . . . 4 (ω ≼ 𝐴𝐴𝐴)
8988adantr 484 . . 3 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → 𝐴𝐴)
9086, 89eqbrtrd 5075 . 2 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
9184, 90pm2.61dan 813 1 (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wex 1787  wcel 2110  Vcvv 3408  cdif 3863  cun 3864  cin 3865  wss 3866  c0 4237  {csn 4541   class class class wbr 5053  ccnv 5550  ran crn 5552  cres 5553  cima 5554  Fun wfun 6374   Fn wfn 6375  wf 6376  1-1wf1 6377  1-1-ontowf1o 6379  cfv 6380  ωcom 7644  cen 8623  cdom 8624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-er 8391  df-en 8627  df-dom 8628
This theorem is referenced by:  infdiffi  9273  infdju1  9803  infpss  9831
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